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arXiv: 1010.2127

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Large solutions of elliptic systems of second order and applications to the biharmonic equation

Marie-Françoise Bidaut-Véron 1, Marta Garcia-Huidobro 2, Cecilia Yarur 3

(2010-10-08)

In this work we study the nonnegative solutions of the elliptic system Δu=|x|^{a}v^{δ}, Δv=|x|^{b}u^{μ} in the superlinear case μδ>1, which blow up near the boundary of a domain of R^{N}, or at one isolated point. In the radial case we give the precise behavior of the large solutions near the boundary in any dimension N. We also show the existence of infinitely many solutions blowing up at 0. Furthermore, we show that there exists a global positive solution in R^{N}\{0}, large at 0, and we describe its behavior. We apply the results to the sign changing solutions of the biharmonic equation Δ²u=|x|^{b}|u|^{μ}. Our results are based on a new dynamical approach of the radial system by means of a quadratic system of order 4, combined with nonradial upper estimates.

1:	Laboratoire de Mathématiques et Physique Théorique (LMPT)
	CNRS : UMR6083 – Université François Rabelais - Tours
2:	Departamento de Matematicas PUC
	Pontificia Universidad Católica de Chile
3:	Departamento de Matematicas y CC (Departamento de Matematicas y CC)
	Universidad de Santiago de Chile

Subject

:

Mathematics/Analysis of PDEs

Keyword(s): Semilinear elliptic systems – Boundary blow-up – Keller-Osserman estimates – Asymptotic behavior – Biharmonic equation

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From: Marie-Françoise Bidaut-Véron <>

Submitted on: Monday, 11 October 2010 09:02:04

Updated on: Monday, 11 October 2010 17:02:08